Fundamental solutions and complex cotangent line fields. (English) Zbl 1283.32002

Webster, Sidney M.

Ill. J. Math. 56, No. 1, 251-263 (2012).

Author’s abstract: We consider a fundamental solution for the \(\overline{\partial}\)-operator on a complex \(n\)-manifold, which is given by an \((n,n-1)\)-form of the Cauchy-Leray type \(\Theta=\theta\wedge(\overline{\partial}\theta)^{n-1}\), where \(\theta \) is a suitable \((1,0)\)-form. On the open submanifold \(M^{n}\) where \(\theta\) is smooth and nonzero, its multiples generate a complex line sub-bundle \(E\subset T^{*}_{(1,0)}M\), which we assume to satisfy a certain integrability condition. To such an \(E\) we attach a global holomorphic invariant, in the form of a complex Godbillon-Vey \(\partial \)-cohomology class, provided a certain primary obstruction class vanishes. If \(\theta \) is also Levi nondegenerate, in that \(\Theta\neq0\), then it determines an invariant connection on the hyperplane bundle given by \(\theta=0\). This provides \(\theta \) formally with a complete system of local holomorphic invariants.

Reviewer: Jean Ruppenthal (Wuppertal)

MSC:

32V40	Real submanifolds in complex manifolds
32A26	Integral representations, constructed kernels (e.g., Cauchy, Fantappiè-type kernels)
53A55	Differential invariants (local theory), geometric objects

Keywords:

Cauchy-Fantappiè-Leray kernels; hyperplane bundles; integrability conditions